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Newton's method is a method for approximating the value of the roots of a function that cannot be solved for algebraically. Given the function f(x) and an estimate value for the root x0, the first approximation is

The second is

and in general

The more times this process is repeated, the better the approximation will be.

For best results, Newton's method should be initiated where is sufficiently large, is sufficiently small, and is close to the intended root. This will help avoid certain situations involving unexpected roots, or non-converging situation.

## Example

Suppose we are given the function

We will start with the approximation x0 = -0.5. The first approximation will be

The second will be

Plugging this into the original equation, we get

The more approximations we make, the closer to zero the function will become.

## Damped Newton's Method

To help improve convergence, newton's method may be dampened with a constant α from (0,1].

Ideally, the each value of α should have the next iteration get as close to the root as possible. Only possible method of determining α is the Bank-Rose algorithm.[1]